The 23-year-old Colombia forward, who scored a tournament-leading six goals at the World Cup in Brazil, was presented to fans at the Santiago Bernabeu Stadium yesterday afternoon.
Rodriguez passed a medical examination earlier at the Sanitas Clinic in the northern suburb of La Moraleja.
Madrid agreed to pay Monaco 63 million pounds for Rodriguez who would join the squad on Aug. 1.
Rodriguez, who had joined Monaco from FC Porto last year, is Madrid’s second signing after Germany midfielder Toni Kroos joined Thursday from Bayern Munich.
Rodriguez’s fee is fourth highest in transfer history following Gareth Bale (86 m pounds), Cristiano Ronaldo (80 million) and Luis Suarez (75 million).
Meanwhile The Brazilian Football Confederation (CBF) yesterday appointed Dunga as coach, replacing Luiz Felipe Scolari, whose contract was not renewed after the hosts’ World Cup semi-final thrashing by Germany.
The 50-year-old Dunga, Brazil’s 1994 World-Cup winning skipper, takes the reins for a second time having led the Selecao at the 2010 tournament.
“I am immensely happy – thank you for your confidence in me,” said Dunga, who in his playing days was a tough-tackling midfielder with Serie A side Fiorentina and Germany’s Stuttgart, and who also had a spell in Japan with Jubilo Iwata.
“The fans are very down right now but they are right behind the team,” Dunga continued.
“I am not here to sell a dream, we must get down to work.”
His no nonsense style was decried by some but his 91 caps showed the esteem in which he was held and that respect has endured, CBF president Jose Maria Marin said.
Dunga was fired after a quarter-final loss to Holland in South Africa four years ago but previously landed both the Copa America title and the Confederations Cup having replaced Carlos Alberto Parreira as coach in 2006 despite having no dugout experience.
Brazil suffered one of the biggest upsets in World Cup history at home this month after they lost their semi-final 7-1 to eventual champions Germany.